SECTION CHECK IN – 601 DIMENSIONAL ANALYSIS QUESTIONS

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AS and A Level Further Mathematics A Check In 6.01 Dimensional analysis

Section Check In – 6.01 Dimensional Analysis

Questions

1. The dimensions of work are:

(A) and it is a scalar; (B) and it is a vector;

(E) and it is a vector.

2. From the following list of dimensional formula select the correct one for each quantity in the table and write it in the appropriate cell.

(A)	acceleration
(B)	volume
(C)	force
(D)	density
(E)	pressure
(F)	energy

3. Use the formula for arc length in a circle to show that the dimension of “angle” is 1

4. Using the usual symbols for mass, height, velocity and force, which of the following expressions for the energy E of a system are not dimensionally consistent with energy?

SECTION CHECK IN – 601 DIMENSIONAL ANALYSIS QUESTIONS

5.* Identify which of the following formula are dimensionally inconsistent, given that v and u are velocities; a and g are accelerations; s and h are distances; m is mass; F is force; p is pressure; V is volume; is density; t is time and is the coefficient of friction.

SECTION CHECK IN – 601 DIMENSIONAL ANALYSIS QUESTIONS

6. A small mass is suspended from a long thread to form a simple pendulum.

The period, T, of the oscillation will depend on the mass, m, the length of the

thread, l, and the acceleration, g, due to gravity.

So where k is a constant and x, y and z are numbers.

Use dimensional analysis to find x, y and z, and hence derive the standard equation for the period T of a simple pendulum in terms of k, m, l and g.

7.* (i) Young’s modulus is the name given to the modulus of elasticity of a wire, and is

defined as . Given that strain is the ratio of 2 lengths and stress is , derive the dimensions of Young’s modulus.

(ii) When a stretched wire vibrates, the frequency (f) of the vibration depends on several factors.

Barry thinks that the frequency will depend on the mass per unit length, (m), the length of the vibrating element of the wire (l) and the force (F) used to stretch the wire.

However,Tom thinks that the density () should be used instead of the mass per unit length.

Formulate alternative models for the frequency using Barry’s and Tom’s assumptions.

8. When an object is falling through the atmosphere towards the ground it is subject to two external forces:(i) the gravitational force and (ii) the air resistance or drag of the object.

The net external force, F, is equal to the difference between the weight W and the drag D. When W = D, the object will be travelling at its terminal velocity, i.e. zero acceleration.

The magnitude of the drag, D, depends upon a dimensionless drag coefficient, , the density of the air, , the square of velocity, V, and the cross sectional area of the object, A

Thus, the terminal velocity, , can be calculated as .

Verify that this relationship is dimensionally correct.

9.* The velocity of sound waves through any material depends on (i) its density,, and (ii) its modulus of elasticity, E.

Given that the dimensions of the modulus of elasticity are , use dimensional analysis to suggest a relationship between the velocity of the sound waves, the density and the modulus of elasticity.

10. A water container is filled to a depth h. When a small hole is drilled in the bottom of the container it takes t seconds for the water to run out. Assuming that t depends on h and the acceleration due to gravity, g, formulate a model for t.